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 A234470 Number of ways to write n = k + m with k > 0 and m > 2 such that p(k + phi(m)/2) is prime, where p(.) is the partition function (A000041) and phi(.) is Euler's totient function. 17
 0, 0, 0, 1, 2, 3, 4, 5, 5, 4, 4, 4, 2, 2, 3, 5, 4, 2, 4, 2, 3, 2, 3, 2, 3, 1, 0, 3, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 4, 2, 1, 4, 2, 1, 2, 3, 3, 3, 1, 0, 4, 2, 4, 1, 1, 2, 2, 3, 2, 2, 0, 2, 2, 1, 2, 2, 1, 1, 2, 2, 4, 2, 1, 0, 1, 3, 1, 0, 2, 4, 3, 1, 6, 2, 2, 1, 2, 4, 3, 1, 2, 6, 2, 3, 2, 2, 2, 2, 3, 3 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Conjecture: a(n) > 0 if n > 3 is not among 27, 34, 50, 61, 74, 78, 115, 120, 123, 127. This implies that there are infinitely many primes in the range of the partition function p(n). LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..6500 Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014 EXAMPLE a(26) = 1 since 26 = 2 + 24 with p(2 + phi(24)/2) = p(6) = 11 prime. a(54) = 1 since 54 = 27 + 27 with p(27 + phi(27)/2) = p(36) = 17977 prime. a(73) = 1 since 73 = 1 + 72 with p(1 + phi(72)/2) = p(36) = 17977 prime. a(110) = 1 since 110 = 65 + 45 with p(65 + phi(45)/2) = p(77) = 10619863 prime. a(150) = 1 since 150 = 123 + 27 with p(123 + phi(27)/2) = p(132) = 6620830889 prime. a(170) = 1 since 170 = 167 + 3 with p(167 + phi(3)/2) = p(168) = 228204732751 prime. MATHEMATICA f[n_, k_]:=PartitionsP[k+EulerPhi[n-k]/2] a[n_]:=Sum[If[PrimeQ[f[n, k]], 1, 0], {k, 1, n-3}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000010, A000040, A000041, A049575, A232504, A233307, A233346, A233918, A234200, A234246, A234309, A234337, A234344, A234347, A234359, A234360, A234451, A234475 Sequence in context: A301534 A101916 A100771 * A113771 A131845 A062186 Adjacent sequences:  A234467 A234468 A234469 * A234471 A234472 A234473 KEYWORD nonn AUTHOR Zhi-Wei Sun, Dec 26 2013 STATUS approved

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Last modified August 22 20:23 EDT 2019. Contains 326182 sequences. (Running on oeis4.)